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DEPARTM ENT OF MATHEMATICS

PALAMURU UNIVERSITYM.Sc. Mathematics

MM IOI

Unit I

Automaphisms- Conjugacy and

104to128of [1] )

Semester I

G-sets- Normal series solvable groups- Nilpotent groups. (pages

Itit lt

Structure theorems of groups: Direct product- Finitly generated abelian groups- lnvariants of afinite abelian group- sylow's theorems- Groups of orders p2,pQ . (pages 13g to 155)

lmuldeals and homomsphism- Sum and direct sum of ideals, Maximal and prime ideals- Nilpotentand nil ideals- Zorn's lemma (pages L79 to ZLLI.

Unique factorization domains - Principal ideal domains- Euclidean domains- polynomial ringsover UFD- Rings of traction.(Pages 7L2 to 228l.

Text Books:

[{ Basic Abstract Algebra by P.B. Bhattacharya, S.K. Jain and S.R. Nagpanl.

: 1l Topics in Algebra by l.N. Herstein.

Paper I

MM 151

M.Sc (Mathematics)

Algebra

PaPer IPractical Questions

Semester I

LAfinitegroupGhavingmorethantwoelementsandwiththeconditionthatx,*eforsomexeGmusthavenontrivialautomorphism.

2. (i) Let G be a group Define a* x = ox) a'xeG then the set G is a G-set

(ii) Let G be a group Define a* x -- axo-' a'xe G then G is a G-set'

3.AnabeliangroupGhasacompositionseriesifandonlyifGisfinite4.Findthenu,u",ofdifferentnecklaceswithpbeadspprimewherethebeads

can have anY of r different colours

5.lfGisafinitecyclicgroupofordernthentheorderofAutG,thegroupofautomorphisms of G ,is Q(n),where / is Eurer's function.

6.lfeachelement+eofafinitegroupGisoforder2then|G|=2nand

G o CtxCrx "" "xC,where C ' arecyclic and lC'l = 2'

7.(i)Showthatthegroup:Gisadirectsumoftt={05\andf={614691( ' 'l ..nnot be written as the direct sum of two

(ii) Show that the group [O ' *r1 uott"-'

SubgrouPs of order 2'

8. (i) Find the non isomorphic abelian groups of order 350

(ii)lfagroupoforderp,containsexactlyonesubgroupeachoforders

p,pz , _ __p,-r then it is cYclic.

.9. Prove that there are no simple groups of orders 63' 56' and 36

l0.LetGbeagioupoforderl08.Showthatthereexistsanormalsubgroupof

11. (i) :::x;JJ*mutative Ring wirth unitv. suppose R has no nontrivial ideals 'Prove

that R is a field' Z

(ii) Find allideals in Z and'n f o)

Board of Studlss in ldat&*n attr"

i12. (i)The only Homomorphism from the ring of integers ZtoZare the identity and

Zero Mappings.

(ii) Show that any nonzero homomorphism of a field F into a ring R is one-one.

13. For any tow ideals A and B in a Ring R (i) # " #,.1,A*B A+B A+B Btlll-=-x ! x,--, A(\B A B AaB *ln particular if R = A+ B then

.R .RR r-x-.AnB AB14. Let R be a commutative ring with unity in which each ideat is prime then R is a

field

L5. Let R be a Boolean ring then each prime idear p + R is maximal.16. The commutative integra I domain R = {a + [J-- 5 I a,b e Z] is nota U FD.

17. (i) The ring of integers Zis a Euclidean domain(ii)The Ring of Gausion lntegers ft={m+nl-l /m,neZ}is a Euclidean domain

18. (i) Prove that 2+.GS is irreducible but not prime in Z1^{j.,(ii) Show that I +2J-i and3 are relatively prime in 21.{-51

19. Let R be a Euclidean domain . prove the following(i) tf b*0tnen6@)<fi1b1

(ii) lf a and b are associates then O@) = O@)

(iii) lt alb and {(a)=O(b) then a and b are associates

20. Prove that every nonzero prime idear in a Euclidean domain is maximal.

,Aay-v!q.U itr4ll. tBoard of Si .. ,l

ffiiHffiy.rlarrcaa'a

i* DEPARTM ENT OF MATHEMATICS

Semester IMM - 102

,er

PALAMURU UNIVERSITY

M.Sc. Mathematics

AnalYsis /

PaPer-II

Metric spaces- Compact sets- Perfect sets- Connected sets

Limits of functions- Continuous functions- Continuity and compactness Continuity and

connectedness- Discontinuities - Monotone functions'

Unitlll

Rieman- ste,tjes integrar- Definition and Existence of the lntegral- Properties of the integral-

lntegration of *tto' vatued functions- Rectifiable waves'

Unit-lV

Sequencesandseriesoffunctions:Uniformconvergence-Uniformconvergenceandcontinuity- Uniform convergence .nr',na"rr.,ion- uniform convergence and differentiation-

Approximatt"" "" continuous tunction by1 sequence of polynomials'

;, ;ffis of MathematicarAnalysis [3,d edition) (chapters 2,4,6) Bv warter Rudin'

r-' Mc Graw-Hill lnternation Edition'

v[iilrtrlarr,Board of Stiriies in h{athemrtlaOmrnfa UniversltY,'t*dcrabad-sfxlrrn

MM 152

M.Sc. Mathematics

Analysis

Paper -IIIPractical Questions

Semester -

is the largest open set

l. construct a bounded set of real numbers with exactly three limit points2. Suppose Et is the set of all limit points of E. Prove that Er is closed also prove that E and

have the same limit points.3. Let E0 demote the set of all interior points of a set E. prove that E0

contained in E Also prove that E is open if and only if E = E04. Let Xbe an infinite set.Forp e X, Q e X define

db,q)={;'{rrr.=r,

Prove that this is a metric, which subsets of the resulting metric space are open, whichareclosed? Which are compact?

5' i) If A and B are disjoint closed sets in some metric space X, prove that they are separatedii) Prove the same for disjoint open setsiii)Fixa p e X arrd d > o, Let A= { q e X :d(p,q) < 6 |

md B ={q eX d(p,q)>d} provethatAandBareseparated.

6- i) Suppose f is a real function on R which satisfies ltgV]u + h) - .f (x - h)= ,] fo, everyx e RDoes this imply that f is continuous? Explain

ii) Let f be a continuous real function on a metric space X,let z(D: {p . x : .f (p) =o}prove that z (f) is closed.

7. If f is a continuous mapping of a metric space X into a metric space y .prove that

, f (E). f @) for every set E cX

8' Letf andgbe continuousmapping of ametric spaceX into amQtric space yLetEbe adensesubset of X. Prove that

i) (E) is dense in f(X)ii) Ifg(p):f(p) Vp € E,provethatg(p):f(p) Vp eX

9. Suppose f is a uniformly continuous mapping of a metric space X into a metric space y and {X,n) is a couchy sequence in X prove that {f(X,n)} is cauchy sequence in y

Board cl .,;l iu ldathcmaU*O:nla Univcrsity,Ufaartad-500orn

Board of Stu,-ii,::. rl Mathemattcr

t$-LetI:[0,1]betheclosedunitinterval,supposefisacontinuousmappingof fintol'Prove

that f(x) = x for at least one x

ll.Suppose AincreasesonIa,b],a<xocb, aiscontinuousatxs'f(x6)= 1 andf(x):0ifx *xo

b

.Pro;e that f eR(aland lf da =O

12.Supposefzgandf ir"o,,,i""ousonIa,b] and '!f4>a*=0'Provethatf(x):0 V x€[a'b]

13.If (x): lor 0 according as x is rational or rrot .pi*" thatf 4.R on I a , b] for any a,b,€R with

a+Also prove that f e R( a) on I a , b] with respect to any monotonically increasing function

a onIa,b]l4- Srrypose f is abounded real function on I a, b] and f€n on I a, b]' Does it follow that f e R'!

Does the answer change if we assume tfrat fCR?

15. Srrypose y,and y,arethecurves in the complex plane defined on [0,2 zt)by r r(t): ei. , Tz(t) = e2',

Show that the two curves have the same range

AlsoShowthat/landy,arerectifiableandfindthecurvelenlthofy,andy,

16. Discuss the uniform conversance of the sequence of functions {fn} where

f.(x)= $ xreal, n =!,2,3""4n

17. Give an example of a series of continuous functions whose sum function may be discontinuous'

18. Discuss the uniform conversance ofthe sequence

f'E)=-J-x ) 0' n = l'2'3 "'l+nx

19. Give an example of a sequence of functions such that

firn [f, .* [t^7,

prove that a sequence {fn} converse to f with respect to the metric of c(x) if and only if f" -+ f

uniformlY on X :

Ornunia Universlty,{ydmabad-s00Otr.

20.

€Y

\.

DEPARTMENT OF MATHEMATICS

PALAMURU UNIVERSITY

B'MM - 103

M.Sc. fMathematics)

Mathematical Methods /Paper- III

Unit IExistence and Uniqueness of solution of + = l(x,y).The method of successive approximation-

ctx

Picard's theorem- Sturm-Liouville's boundary value problem.Partial Differential Equations: Origins of first-order PDES-Linear equation of first-order-Lagrange's method of solving PDE of Pp+Qq - R - Non-Linear PDE of order one-Charpitmethod- Linear PDES with constant coefficients.

Unit II

Partial Differential Equations of order two with variable coeffrcients- Canonical formClassification of second order PDE- separation of variable method solving the one-dimensionalHeat equation and Wave equation- Laplace equation.

Unit III

Power Series solution of O.D.E. - Ordinary and Singular points- Series solution about an ordinarypoint -Series solution about Singular point-Frobenius Method.Laeendre Polynomials: Lengendre's equation and its solution- Lengendre Polynomial and itsproperties- Generating function-Orthogonal properties- Recurrance relations- Laplace's definiteintegrals for P, (x)- Rodrigue's formula.

Unit-IVBessels Funstions: Bessel's equation and its solution- Bessel function of the first kind and itsproperties- Recurrence Relations- Generating function- Orthogonality properties.Hermite Polynomials: Hermite's equation and its solution- Hermite polynomial and its properties-Generating function- Altemative expressions (Rodrigue's formula)- Orthogonality properties-Recurrence Relations.

Text Books:

[1] "Elements of Partial Differential Equations", .By Ian Sneddon, Mc.Graw-HillIntemational Edition.

[2] "Text book of Ordinary Differential Equation", By S.G.Deo, V. LakshmiKantham, V.Raghavendra, Tata Mc.Graw Hill Pub. Company Ltd.

[3] "Ordinary and Partial Differential Equations", By M.D. Raisingania,S. Chand Company Ltd., New Delhi.

Semester I

,sr,rr*tl, uoil: t'r b8atb

Y*1:;';,:l:1..?''- '{etabao--'

'h,t':

tt

M.Sc. MathematicsMathematical Methods

PaPer IIIPractical Questions

vnfi',tsl Semester I

1. Compute the first three successive approximations for the solution of the initialvalue

dxProblem fr=*', x(0)=1'

2. Solve yp =2W+logq.

3. Solve yzp + zxq = ry with usual notations'

4. Explain Strum-Liouille's boundary value problems'

s. crassiff the equation #.^#. ^#-**2ff=o'

6. Solve r +t +2s =0 withthe usual notations'

Find the particular integral of the equation (D2 - D)Z = e2'*v

Solve in series the equation xy + y - ! =0 '

g. Solve y' - ! = x using power series method'

10. Solve the Froenius method x' y" * 2*' y' - 2y = 0 '

I 1. Solve in series ZxY' + 6Y' + Y = g'

12. Prove that J-,(x) = (-l)'J,(x) where n is an integer'

13. Prove that x.4(r) = nJ,(x) - il,u(x)'

14. Prove thal H,(-x1 = (-l)'Il,(x)'

7.

8.

es ln tt{athcmatlg

ersitY,H:"{i}

15. Show that I/r,*,(0) = 0.

. 16. Show that (2n+l)x\(x)=(n*l)p,.,(x) +np,_,(x).

17. Solve with u(x,O) = 4e-' using separation of variable method.

18. Find the surface passing through the parabolassatisfying the equation xr + zp - O.

Z =0, l'=4m and Z =1, !, =-4ax and

19. Find the surface satisfying t = 6x2y containing two lines Y=0:zandy=2:2.

20. Reduse the equation x,r - yrt + px - (ty = x, inthe canonical form.

Au 0u

Ax Ay'

DEPARTMENT OF MATHEMATICSPALAMURU rrNrvERsriv

M.Sc. MathematicsSemester I

MM104

uharmaD,'^ talirepiflppBoard ol'Sr ' 'i "

O.,*a[iA t, ;',lYrt's]ffrt$attrabad-sfrrrH

Elementary Number Theorypaper_ IV

UNIT.I

The Division Argorithm- Number patterns- prime and composite Numbers_Fibonacci and Lucas' numbers- Fermat Numbers- GCD-The Euclidean Algorithm_The Fundamentar Theorem of Arithmetic- LCM- rir"u, oioprrrrtine Equations

UNIT.II

congruences- Linear congruences- The r..orraf Rho Factoring Method- DivisibilityTests- Modular Designs- check Digits- The chinsse Remainter Theorem- GeneralLinear Systems- 2X2 Systems

UNIT-III

wilson's Theorem- Fermat's Little Theorem- pseudo primes_ Euler,s Theorem-Euler's Phi function Revisisted- The Tau and sigma Functions_ perfect Numbers_Mersenne Primes- The Mobius Function

UNIT,IV

The order of a Positive Integer- Primality Tests- primitive Roots for primes-composites with primitive. rooti- The Algebra or rnai..r] erJaruri, Residues- TheLegendre Symbol- euadratic Reciprociry-

-fhe Jacobi Symboi

Text Book : Thomas Koshy ,Erementary Number Theory with Apprications

Fia' .f

iElementary Number theory

Practical QuestionsMM154 Paper IV

1Find ife positive integer d if la, a * ll = 132.

2Find the twin primes p and 4 such that Ur, ql = 323.

3The LDE ax + W: c is solvable if and oriy if dlc, where d: (a, b). If xo, yo is aparticular solution of the LDE, then all its solutions are given by

^. (1), and ':*- (1),

where, is an arbitrary integer.

4Solve the LDE 107 6x -t 207 6y : 307 6 by Euler's method.

5End the general solution of each LDEb+3y:4l?t+l3y=14

6Dermine the number of incongruent solutions of each

fucongruence.

l2.r: l8 (mod 15)

28u= ll9 (mod 9l)49x=94 (mod 36)

7lhirycongruences, solve each LDE.

3r+4r:5lst+ 2l): 39

IUsing the Pollard rho method, factor the integer 3893.

IProrrc that the digital root of the product of twin primes, other than 3 and 5, is 8.

a10:Using the CRI, solve Sun-Tsu's plzzle:

.r= 1 (mod3), x=2 (mod5), and x: 3 (mod7)

Semester I

A'#bfbi\t*\,

lbf,ll+tHT

itffifficrnatl':'

77Prove each, where p is a prime.

Ietp be odd. Then}@ - 3)!: -l (modp).@ - Y!9 - 2)...@ - k) - (-Dkkt (mod p), where1.k.1.12Find the remainder *1nn 24tea7 is divided by 17.

13I*t p be any odd prime and a any nonnegative integer.Prove the following.

1P-t S2n-t +... + (p - l)p-l : -l (modp)tP +?? +...+ (p - l)r:6 1^*r,

74

Verify each.

(12+lrt7:1217 *1517 (mod l7)(16+2D23 :1623 +2123 (mod 23)

15

Find the remainder when z45te,0 is divided by l g.

76

Evaluate (-4141) and (-9183).

t7

Verify that 99731(24e86 + l).

1B

Prove that there are infinitely many primes ofthe form 4n * l.L9

Show that l! + 2! + 3! * ...* nl is never a square, where n > 3.

20

Prrove that there are infinitely many primes of the form 10ft _ l

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MM -201

DEPARTMENT OF MATHEMATICS

PALAMURU I.JNIVERSITY

M.Sc. (Mathematics)

Advanced Algebra

Paper I

Semester II

Unit I

Algebraic extensions of fields: lrreducible polynomials and Eisenstein criterion- Adjunction ofroots- Algebraic extensions-Algebraically closed fields (Pages 281to 299)

Unit ll

Normal and separable extensions: Splitting fields- Normal. extensions- Multiple roots- Finite fields-

Separable extensions (Pages 300 to 32L)

Unit llt

Galois theory: Automorphism groups and fixed fields- Fundamental theorem of Galois theory-

Fundamental theorem of Algebra (Pages 322 to 339)

Applications of Galoes theory to classical problems: Roots of unity and cyclotomic polynomials-

Cyclic extensions- Polynomials solvable by radicals- Ruler and Compass constructions. (Pages

340-364)

Text Books:

t.

[1] Basic Abstract Algebra- S.K. Jain, P.B. Bhattacharya, S.R. Nagpaul.

Reference Book:Topics in AlgrbraByl. N. Herstein

MM 25'1

M.Sc. Mathematics

Advanced Algebra

Paper IPractical Questions

1. (i) lr@)=l+x+.....+xp-r is irreducible overQ. Where p is a prime.

(ii) Showthat x3 +3x+2.flQ)is irreducibte overthe fietd *.\7)' ', \7)

2. Show that the following polynomials are irreducible over e(i) x3 -x-t (ii) xa -3x2 +9 (iii) xa +8

3. Show that there exists an extension of E ,f 6

with nine elements having all

theroots of x2 -x4e*G)(r/

4. (i) Show that there is an extension E of R having all the roots of I + x2

(ii) Let l(x)ef(x)for t=L,2,..... . .mthenthereexistsanextension E of

F in which each polynomial has root

5. Show that Ji ond .11 are algebraic over Q and find the degree of O[z)over eano 9('.6) over Q.

(iii) Find a suitabte number a such that e(Ji,Ji)= efot.6. Show that the degree of the extension of the splitting field of x' -2 e e@) is 6

7. Let p be a prime then /(x) = xP -l e Q@) has a splitting field Q@) where d, *land ap = l.Also lg@:el= p-l

8. Show that the splitting field'of f(*) = xn -2. QQ) over Q *g1ri,i)and itsdegree of extension is 8

9., lf the multiplicative group F' of non zero elements-of a field F is cyclicthen F is

Finite

10. The group of automorphisms of a field F with p" elements is cyclic of order n and

generated by / where 0@) = xP, x e F

1t.The group Cp@l where ds =l and a+ I is isomorphic to the cyctic group of'o' ,

Semester II

order4 dffi' ;I l'J:[,Mra

t h ernp u&tlvdcrabad_r.r'#

12. Let B = g1\l-2,ot)whereco3 =l,o *lleto, be the identity automorphism of E and

Let o, be an automorphism of E such that or(ar) = 02 and or({i) = o$l2)'lf

G = {ot,or} then Eo = Q\llat')fg. tfr,f(x) e F(x) has r distinct roots in its splitting field E over F then the Galois

sroup ,(#)* ,(x) is a subgroup of the symmetric group s".

L4. The Gatois group of xa -z e Q(x) is the octic group'

15. The Gatois group of xo + | e Q@) is Klein four group

( z\'!6. $r@)and xs -1 have the same Galois group namely

t Ol = {1,3,5,7\,the Klein's

four grouP.

t7.lt afield F contains a primitive nthroot of unity then the characteristic of F is Zero

or a prime P that does not divide n

1g. Show that the following poJynomials are not solvable by radicals over Q

1i1 ,'-t0r5 +l5x+5 (ii) x' -gx+3 (iii) xs -4x+2

19. lt is impossible to construct a cube with a volume equal to twice the volume of a

given cube by using ruler and compass only.

20. A regular n-gon is constructible if and only 'rt 0b) is a power of 2. (equivalently the

angle ? irConstructible.)n

$rrrrnm3.[ilrd of Studlec ln l{athernatte*hnfe Unhersity.tlvdaahnd*{r'Yrfln+

rlr

MJfi_M2I

D E PARTM E NT OE,MATH E MAT! CS

PAtAMffiU:.trM.Sc. Mathematics

Advanoed AnalysisPaper II

Semeste II

Unit I

Algebra of sets- Borel sets- Outer measure- Measurable sets and Lebesgue measure- A non-

measurable set- Measurable functions- Little word's three principles.

Unit ll

The Rieman integral- The Lebesgue integral of a bounded function over a set of finite measure-

The integral of a non-negative function- The general Lebesgue integral.

Unit tll

Convergence in measure- Differentiation of a monotone functions- Functions of bounded variation.

Unit-lV

Differentiation of an integral- Absolute continuity- The Le-spaces- The Minkowski and Holde/s

inequalities- Convergence and completeness.

f,:t Bools:ll] Real Analysis (3'd EditionllChapters 3, 4, 5 )

., by

H. L Royden

Pearson Education (Low Price Edition)

-,ll.rllllli'Ur,ffi

"r itudieqftr N{ath e rnaurct

116-'iE!tC-'#

c!.F \'rD E P ART M E NT O F M ATH E fVI AT I CS

PALAMURU UNIVERSITYtvt.Jc. lvluutetilultcs

Mtvll0l

UNIT-I

Regions in the Complex Plane -Functions of a Oomplex Variable - Mappings -Mappings by the

Exponential Function- Limits - Limits Involving the Poinl ut Infinity - Continuitl, -i lDerivatives -

Cauchv-Riemann Ecluations -Sufficient Conditions.for Dffirentiabilin'- Analt'tic Function.s

Harmonic Ftmctions - LIniquell'Determine tl ..1nttl.l'tic' Funt'tittn,s - ReJleclion Principle - The

Exponential Function -The Logarithmic Function -Some ldentities Involving Logtlithnrs -

C omplex Exponents -'' Tri gonome tric Func tions -l{y'pe rbol ic Func tion s

UNIT.I|

Derivotives of Functions w(f) - Definite Integrals q/'Functions w(t) - Contours -Conlour

Integrals - Some Examples -Examples with Branch (uls - Upper Bounds.for Moduli o_[

Conlttur Integrals -Anli derivctlives -Cuuchy-Goursul Theorem -Sintply Connecled Domuins-lvfuhiplv Connected Domains -Cauclry Integral Formulo -An Extension of the Cauchy IntegrulFormula -Liouville's Theorem and lhe Fwrdamenlal Theorem oJAlgebra -Ju[aximum fuIodulus

Principle

uNtr-lll

Contergence of Sequences - Convergence o.f Series -Taylor Series -Latn'enl Series -Absolute antlL'ni.lorrn Convergence of Pox'er Series- C'onlinuity' of Suns of Pow'er Series - Integration andDi-tferentiotion of Power Series - Uniqueness of Series Representations-l.solated Singular Poittts- Residues - Cauchy's Residue Theorem - Residue at Infinity - The Three Type.s o-f Isoluted

Singulor Points - Residues at Poles -i )Examples -, 'Zeros of Analytic Functions -Zeros and Poles

-Beha,ior of Functions Near Isolated Singular Points

UNTT.IV

Evaluation o_l'Improper Integrals -Improper lntegralsfrom Fourier Analysis - Jordan's Lemma -

Indented Paths - - Definite Integrals In'volving Sines and Cosines - Argument Principle 'Rouche"s Theorem -Linear Transformations -The Transformation v, : l/z - Mappings by l/z -Linear Fractional Transformations -An Implicit Form - 'Mapping.s cf the Upper Half Plane

Texr: James Ward Brou,n, Ruel V Cfutrchill, Complex Variables with applications

SC,nrcitcr'l lrl Corripl*:x Arral-vsisPoper-l

+.I fvl.r{l:Senrr'ster Lll

Unit-l ' ' ":'1r';:::rr:,:' l,"t:li,lrt:ir.:i

NoRlllED I-INE.{R SPi\clES: Dcllniri"s:1,i,*iirnl# huperries. subspacc.. crosctrsuhspar:c. Iiinite f)ittrension:rl N.r'r*ctt''riinti.rspirccs,,|,.i s,,trrpr.:*.. qru,i.,.,r s1:,cr,s.(', rlttlrlctiorr 1l \r rrrrrt (l Slr,,..*.

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HII'UERT sPA( ES: lnncr Pnxlttct s;:ltcu' Hritrt'n . ip.1.,". ( .ucrrv-B.nl,akovsk\-schr.irrrz

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Llnir-lll rai.,r.r,.. ..-

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FI]NDA\{ENTAL I-HEoR[i\rs r:c)R BAN;\cH spi\cEs .,\ND ADJotNt- opt]R;\Tot{S tNHILBERT SP;\{:E.S: Clrrsetj Crirph T)rcor.ern. OP311 Ntapping li",;;;r. Bounrlc1 lnvcrseThetxertl. Atl.irint Opcratirrs. Sell-,\iljrrint ()1^*raltrr,s. qlartr,,ri. [r'rrrr. L;nit,ry q)(:rilt(n.s.Projr'cti,,rt (-)l)ct irl (,r\.

Tcxt Br'ok:

A First ('oulse ill Frrnctionirl Analysis-ltabindrariutl sen. Anthcnr prtss ;\n rrnJrri,t .l\Yirnbledt:rr Publishing Cr:rnrpany.

Rel'ert ncc.: '

l-. lntr*dutt,'rry Firnctionrr Anarysis- E Krtyzig- John wirelv and sors. Nerv yo.k.L Fun(ti()nill Arr;rl1,sis. tr.v B.\, I_irrrtrr,., l,a L..ilitiorr.'j' fittxrtltrctiorl to iirPol.luy nntl I\"lrrtlern i\nalvsis- ci.F.sinrnronr- r,lc.(inrrv-Hill ln(crnati'nrrl

Edition.

DEPAR

P x{il'l'Y

N.tM -303 A Si:nre.ster -lII

tlnprr.Il I l.\ l

Lxm-I ,':ttirfitt.',LATTICES: Parri{l Orderi!.!8,'tr,.t{{i9rs;,S.,t[SeJs,- :ionr properties of

'. Lattices - Lattices rs .4igi.birtiE$iiie;ii!,$-.;:..SubiltliCes. Direct producrs and

Hrxrirruorphistu.t - tot,lg,' tP rul;$1f$'1,'.'Li*lrEle. L'olnplelncnled and

dist:ihuti vc latticcs.

tPages I .S-1- I 9?. ,t?ts 397'd l llf ,,:, .',1....r- r,.. ,.t,

t:NI't'- ilBOr)LEAN ALGEBRA: Bwlean Algebr:rs as Lrttices - Bohiur ldtrrtities -rhe switching .Algebra - sub algehra. Dired ProdrcI and hc'nr'rntorphism -Joili irredrcihle ehrnenis - Ati;lm{.lii:li{ni$' r1:- Beole-an fonns and lheirequivalence - mintenn Boclear tor11.15$un1 af prcducls carlonical fonns -*:rtues <ri' Btxrlear erpressions.arq-l:.B.,:v:rfe-*ir 'i'lrnc'tinns - I\'Iinimizet]or: ofBoolc;ur funciians - thc Karnaugh mgp,rEelhcdr

lPages 397 - :l-]t, of l l l) , ,,',

UNI'f- lll ' rr':,:r'rr":r r' '

CR.q,PHS A:$D PL{NAR CRAP!{ ,,: Direercd and undinvc;ed graplx -lrnntorphisn oi grrphs - sr$$rafihrr::eompl+te grif,h - rrrrlfigraphs and

o.eighied grtphs prths srrnpltr ',md- olefiEr*ar-v Prths circuitr

connecredrxss - shon:st paths in'weighted gllphs - Eulerian paths and

ciri"uits - lncomir:g tlegrce antl oulgL{rg rlggree of & vene,\ - Harailtooian

paths an,J circuits - Pla*r graptts -:Euier's ioflnilla fenplanm graphs-

{Page.r li7-119. 168-l.t6itf [?]l

t;Iit1'- I\TREIS AND CUT-SETS: Itl.perties Of trrre: - Equivaleut defin:tions ol trces

- ll.xrletl ttee) - llilfiCv trces - pa{lt leitrgttrs ill luo[ed tees - Preil-r utxJer -Bin';rry search tr*s - Spanniltg trees and CuLseB - lr{inimum spanning uees

(Paees 187-2i3 of[2])

Texl Bnrks:-

tll J P Trirrutrlay und R. lluruhar: Dsurete Matherrutical 511uc1u1e5 rvittl

applrcrrions tn CoilPuter Science, McGra$ Hill Book C'txltpaql_.. - . .- - .Iij C I t_ir, : Elemenrs of Discrcre lrlathernths. Tata lvlrcrarv tlill Publishing

Ct>nrparry Ltd New Delhi. (Second Edition)"

MM _ 3O4A

DEPARTM ENT OF MATHEMATICS

PALAMURU UNIVERSITYM.Sc. (Mathematics)

/OperationS R.r*urrn */

Paper IV A

Semester III

Unit I

Formulation of Linear Programming problems, Graphical solution of Linear Programming problem,

General formulation of Linear Programming problems, Standard and Matrix forms of Linear

Programming problems, Simplex Method, Two-phase method, Big-M method, Method to resolve

degeneracy in Linear Programming problem, Alternative optimal solutions.Solution of simultaneous

equations by simplex Method,..lnverse of a lYlatrix by simplex Method, Concept of Duality in Linear

Programming, Comparison of solutions of the Dual and its primal.

Unit ll

Mathematical formulation of Assignment problem, Reduction theorem, Hungarian Assignment Method,

Travelling salesman problem, Formulation of Travelling Salesman problem as an Assignment problem,

Solution procedure.

Mathematical formulation of Transportation problem, Tabular representation, Methods to find initial

basic feasible solution, North West corner rule, Lowest cost entry method, Vogel's approximation

methods, Optimality test, Method of finding optimal solution, Degeneracy in transportation problem,

Method to resolve degeneracy, Unbalanced transportation problem.

Unit lll

Concept of Dynamic programming, Bellman's principle of optimality, characteristics of Dynamic

programming problem, Backward and Forward recursive approach, Mlnimum path problem, Single

Additive constraint and Multiplicatively separable return, Single Additive constraint and Additively

separable return, Single Multiplicatively constraint and Additively separable return.

Unit-lV

Historical development of CPM/PERT Techniques - Basic steps - Network diagram representation - Rules

for drawing networks - Forward pass and Backward pass computations - Determination of floats -

Determination of critical path - Project evaluation and review techniques updating.

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MN{40r

DEPARTMENT OF MATHEMATICS

PALAMU RU UNIVERSITYM.Sc. Mathernatics

i\ciyiirrr:ed Cornpler z\rtal-v r* /Paper-l

Semester IV

UNIT.I

Entire Functions: Jensen's tbrmula --Functions of finite order- Infinite products Generalities

-Example: the product formula for the sine function -Weierstrass infinite products

Hadamard's factorization theorem

UNIT-II

The Garnnna and Zeta Functions: The gamma function -Analvtic continuation-Further

properties of I- -The zeta function -Functional equation and analytic continuation

uNlT-m

The Zeta Function and Prime Number Theorem: Zercs of the zeta function - Estimates for 1/((s)

- Reduction to the firnctions ry and ry, -Proof of the asynlbtotics for ry, - Note on

interchanging double sums

UNIT-IV

Conformat tvlappings: Conformal equivalence and exarnples -The disc ancl upper half-plane -

Further examples -The Dirichlet problem in a strip -'fhe Schrvarz lemma, a'tutomorphisnrs

of the clisc and upper half-plane-Automorphisms of the disc - Automorphisms of the

l.' Sirr:'lflr:::ltios

f

upper half-plane

Text Book : Elias M Stein, Rami Shakarchi , Complex Analysis

References: Lars V Ahlfors, Complex Analysis

R P Boas, Entire Functions

Lars V Ahlfors, Conformol lnvoriants

,"1. -lr . enni- ..

:'a t!cs:,::'ilnatiCS

.1t '.,

I

#

Adva nced Complex Analysis

Paper-l

MM451

Practical Questions

1

Prove that if lzl < 1, then

(1 +z)(',| +22111 *zo)(1 +28)" =[,'*r',r)=

2

Find the Hadamard Producls for:

(a)d-1,(b) msrz.

3

Prove that for wery z the producl bdow convages, and

ols.rz/2) r:rlz/ 4)cos(z/8) = g aslz/Zk 1 =

4

$row that the equation d - z = 0 has infinitely many solutions in C.

5

Prove Wallis s product formula

n 2'2 4'4 2m '2m

Semester lV

1

r,

snzz

6

Prove that

2 1.3 3.5 (2m - 1) '(2m + 1)

s(s+ 1) '.'(s+ n)

whenevers= 0,-1,-2,

n"n !

^.: llcs, l'natiC6

... .;i.

F(s) = ,,lim

10

#

7

Show that Wallis's product formula can be written as

,lir= nffi12n+'t11t2'd"'l

8

Usethefaci that [(s)f(1 - sl= n/ sinnstoprovethat

lr(1/z+ ,u1= ,ffu, whenever t tR'

9

The Beta function isddined for R{o) > 0 and R{B) > 0 by

:4 B@,F\= [o'$-

qo-1f-1dt.

r(a)r(FI(d prorethar B(o,F)= ffi}i.uo- 1

(b) Shon that 8(o, $ = Jo (@ ou'

Provethat for R{s) > 1,

(s)=# l,- #*.11

Prove as a @nsequence that one has

(((s))2 = t# and ((s)(s- r) = E#n:1

for R(s) > l and R{s- a) > 1, reqedively' Hered(n) equdsthen-um{1

li'iiril-. oi n, "no

o,(r) It the sum of the ath p(t\'\'ers of divisors of n' 'ln

particular, one has so(n) = d(n).

'l+':S-',iatics" '

t'. ,, '

{

rcl]m bounded sequene of complor numbeq

-(-q,Lt nsa=,

wherecn = D-r= ro^&.

when R(s) > 1.

1 - ip(n)<al'!-, ,''

ilObius function defined by

.l iln= 1,

.(-1)r fi n = fi ""p,,, and thep; aredi$inct prime,0 otherwise.

\U!nm| = p@')p(m) whenever n and m are rdativdy prime. [HintEuler produd formula for ((s).1

1[

$row that

15

Prare that the Diridrld series

\: aa/: nsn.1 "

conv€rges for R(s) > 0 and defines a holomorphic function in this half-plane.

Dr(t) =kltt

16

Does there e(i$ a holomorphic srrjmtion

!l if n=1,i 0 otherwise.

from the unit disc to C?

L7

Prove that f(zl= -l(z+

(z= x+ iy:lzl< 1, y> 0J

1/z) is a conformal map

to the upper half-plane.

from the half-disc

/:A=+ '.t .-'.':

1-.+! - '.

{

r"'. .

It

htH thefrmclim u defined by

u(x,Y)= ^ /i+ z\"u \

=/ and u(0, 1) = o

hhnuric in the unit disc and vanisrres on its boundary. Note that u is nothEin D-

t9

-$nthat if t : D(O,R) - C isholomorphic, with lf(z)l s M fotsomeM > 0,

th€n

I rel-rtol l. tztIu,-r<o)re)l'MR

20

Provethat if f :D+ D isanalyticand hastwodi$inct fixed pointg then fistheidentity, that is, f (z) = 7 for all z t:D.

-{t-": - t

1,. /

]\t;b1\r+r'At}trilb*'f,!Jirtrt:'*t*tto#E+bc*u6hdbffiffiHH.n$

H

DEPARTMENT OF MATH EMATICS

PALAMURU UNIVERSITYM.Sc. Mathematics

il{i}I - d02 ,/General Measure TheorY v

Semester IV

Paper- II

Unit I

Measure spaces- Measurable functions- lntegration- General Convergence theorem

Unit ll

Signed measures- The Radon- Nikodym theorem'

Unit lll

Outer measure and measurabriitv- 'f he Extension theorem- The Product measure'

Unit-lV

lnner measure- Extension by sets of measure zero- Caratheodcry outer measure

Text Books:

[1] Real AnalYsis (ChaPters t1',12\

By H.L, RoYden , Pearson Ediation'

,1i1-' " 'r*ties

..,-!i.lr;iivt t:

Y

M.Sc.(Mathematics)

General Measure Theory

Paper ll

MM 452 SemesterlVPractical Questions

1. Let be a non negative Lebesgue measurable function on IR.. For each Lebesgue measurable set of

IR define ( ) = I the Lebesgue integral of over Show that is a measure onthe -algebra of

Lebesgue measurable subsets of IR.

2. Let be a -algebra of subsets of a set and : -) [0, oo]e finitely additive set function. Prove

that is a measure if and onlyif whenever { } is an ascending sequence in then

U )=lim - ( ).

3. Let ( , , b> a measure space and

of Thatis A =[ ]uIdenote the symmetric difference of two subsets and

l. Show that

where and ar real

q m. Find a positive set

A

i. lf and aremeasurableand ( A )=0then ( )= ( )

ii. show that if is complete, and € .Then E if ( A )=0

4. Let ( , b> a measure space and € .Define to be the collection of sets in that are

subsets of and the restriction of to . Show that ( ) is a measure space.

5. Suppose ( , , is)not complete. Let be a subset of a set of measure zero that does not belong

to let = Oon and = . Show that = a.e on which is measurable and is not.

6. Let : --r be a function that is Lebesgue integrable over . For a Lebesgue measurable set

define ( ) = I . Prove that is a signed measure on the Lebesgue measurable space (

Find a Hahn-decomposition of with respect to this signed measure.

7. Let be a measure and and are mutually singular measures on a measurable space ( ,

which = . Show that = 0. Use this to establish uniqueness assertion of th Jordan

decomposition theorem.

8. Show that if is any measurable set then

i. - ( )< ( )<( )

,).

lpr

ii.l ( l)<l l( )

9. Show that if and are two finite signed measures then so is

numbers.Showthatl l= | ll land | + l< I l+ I I

10. !n the question 5 if is a Lebesgue measurable set such that 0 ( (

contained in such that ( ) > 0.Also find a Jordan decomposition of)

riiT:\:.

'naiic-;

il

ll-Let beacollectionofsubsetsofaset and , -[0,oo$easetoffunction.Define.(0)=0andforg*0Define'()_inff,()wheretheinfimumistakenoverallcountablecollections [ ] of sets in that cover Prove that the set function *: ( ) - [0,oo] is an outer

measure called the outer measure induced by

12. Let *: ( ) --r [0,oo] be an outer measure' Let

measurable sets and -U . Show that . (E , { } be a disjoint countable collection of

n)=X .( n ).

13. Show that any measure that is induced by an outer measure is complete'

14.Let beaset, ={0, }nddefine (0)=0and ( )=l'Determinetheoutermeasure *

induced by the set function : - [0,co]nd the -algebra of measurable sets.

15. On the collection 5 of all subsets of define the set function : -+ bY setting ( !o be the

number of integers in . Determine the outer measure . induced by and the algebra of

measOiable sets.

16. Let be a -rlgebra on and a collection of subsets of which is closed under countable

unions and which has the property that each subset of a set in is in ' Show that the collection

=[ i = A , e , i6 ]algebra'

17. i.lf ( ) <-provethat .( ) = ( )- .( ) and

ii. if is a +lgebra then Prove that

'()=inf{(): c , €an}

.()=supt(): c , e ]

18. Suppose is a measure on an algebra of subsets of and is any subset of Jf is the algebra

generated by and and if- and _ are extensions of to such that -( ) = .( ) and

_( ) = -( ). Prove that- and -

are measures on

lg.suppose ={ { n)u( n ): , € }where and areasinproblem18'Provethat

is an algebra of subsets of containing and

20. suppose ( , I a metric space and * be an outer measure on with the property that

'( U ) =,( ) + -( )whenever (, ) > 0. Provethiteveryclosed setis measurablewith

respect to '.

:1'7': /..:I

i.-t.,:,

--,' : ;CS, _,., '.q: -rl'i.rIJv

. , .' -t'l'

-".t,rP{ai

l|'

DEPARTM ENT OF MATHEMATICS

PALAMURU UNIVERSITY

M.Sc. ( Mathematics)PaPer IIIA Semester IV

N{NI - 403A

Inirgral Equalions:Unit I

volterra lrrtegral Equations:,8asic ccncepts ' Relatior,:hip between Linear dlfferential equations

anC \,/clterra !ntegral equations - Resol'.rent Kernel of Volterra lntegral equation Differentiation

cfso=er.es,lventkernels.solr-rtiongflntegr"aloquatinnbvRpsolventKerne|.Themethodofsuccessive approxirnations Ccnvolution type equations - solution of lntegro-dlfferential

equationswiththeaidoftheLaplaceTransformation-Volterraintegralequationofthefirstkind - Euler integrals - Abel's problem - Abel's integral equation and its generalizations'

Unit lt

Fredholm tntegral Equations:Fredholm integral equations of the second kind - Fundamentals -

ThelVlethodofFredholmDeterminants-lteratedKernelsconstructingtheResolventKernelwith

the aid of lterated Kernels - lntegral equations with Degenerated Kernels' Hammerstein type

equation.CharacteristicnumbersandEigenfunctionsanditsproperties.

Green,s function: Construction of Green,s function for ordinary differential equations - Special

caseofGreen,sfunction.UsingGreen.sfunctioninthesolutionofboundaryvalueproblem.

Calculus of Y:rriations:Unit lll

The Method of Variations in Problems with fixed Boundaries:

Definitions of Functionals - variation and lts properties - Euler's'equation - Fundamental

Lemma of Catculus of Variation-The problem of minimum surface of revolution - Minimum

,n.r*, Problem Brachistcchrone Problem - Variational problems involving Several functions -

Functionaldependentonhigherorderderivatives.EulerPoissonequation'

Integral Equations & Calculus of Variation ' '/

iiriri r v

Fr.rnctionai dependent on the Functions of several independent variables - Euler's equations in

1.",.,^ .111r-l^-t'..':l!:hl..r t-/:::i::ti:-:! :::h!:r: i:' r:::::::i:i '' l: t't '';;i'-;'L'-'t ;i i-iL-l;:' CI

''''ivrriution - Hamilton's principle - Lagrange's Equation, Hamilton's equations'

Text Books:

[1] M. KRASNov, A. KISELEV, G. MAKARENKO, Problems and Exercises in lntegral Equations

(1e71)

[2] S. Swarup, lntegral Equations, (2008)

[3] L.EtsGoLTS, Differential Equation and calculus of variations, ]vllR Publishers'

MOSCOW

: ,-... ,

,l

MM453A

M.Sc.(Mathematics)

Integral Equations & Calculus of VariationsPaper lllA

Practical Questions

Semester lV

2.

3.

From an lntegral equation corresponding to the differential equation

y' + xy' +(x' - x)y = xer + l; y(0) =.y'(0) = l; y"(0) = I

Cohvertthedifferential equation yo'+ xy" +(x'- x)y = xe'+ I; with initial conditions

y(0) =.y'(0) = l, .y'(0) = 0; into Volterra's lntegral Equations.

Solve the lntegral Equations

e'(x)+p(x)+ [Sirl,1*-t)A(t)dt+ [Cosh(x-r)E'Q)dt =Coshx ; A(O)= p,(0) = 0.0

Solve the tntegrat Equations !P# = xn ; 01a <l;

with the aid of Resolvent Kernel, find the solution of the lntegralequationrl,

e@) =e..sinx + Fllgll qQ)dti z+cost

Solve the lntegral Equations 0@)- 1'[ur""ost.6(t)dt = +o ^ll- x'Find the characteristic numbers and Eigen function of the lntegral Equations

I

0G) - t !(+s-' togr -of bg*\61t)dt -- o

epplicatiJns of Green's function : Construct Green's function for the homogeneous boundary value

problem y'" (x) = 0; y(0) = .y'(0) = 0; y(l) = y'(l) = 0 .

Applications of Green's function : Solve the Boundary Valuq. problem y"'(-v; = I;y(0) = -rr'(0) = y'(l) =/'(0) = 0.

Applications of Green's function : Solve the Boundary Value problem y' + y = x, ;

Y(o)=Y(r127=s' 'i'nt*"

't*Find the extremals of the functional v[.](x)] = , y

I

7.

10.

11.

12. Test for an extremum the functional v[y(.r)] = I(-, + ),' -21,'y'px; /(0) = l; y(l)) = 2.

(

13.

14.

Find the extremals of the functional v[y(.x)] = !\Af - y" + x'1dx

Determine the extremals of the functional v[y(x)] = I.(tr" + py)dx that satisfies the

boundary conditions y(-l) = y'(-l) = y(l) = y'(l) = 0,tl

15. Find the extremals of the functional v[y(x)] = l{ZW -Zy' + y'' - z'2)dx,Y6

16, Find the extremals of the functional v[y(x)] = '[ ,' + (y')' * -2f]ar

-Jl ' Coshx ).tO -

17. write theostrsradskv equationforthe functionar vfz(x,y))= fl[[*)' -&)'b.

----aili-,,i:lt;,, ..,.,o,-. iti,.ls,) ')lr i!\ 'Y; '-.:',, i;:'' ti::l:

:'--, ,, .. :.,i':-,' ''l'"'.-. ,

'!r' :' -,r..,,iar'.

,, .:1.', ,

''''.'1' ll.VdCrI'Jiir' "

18.

19.

Applications of Hamilton's and Lagrange's equations: Derive the equation of a vibrations of a

Rectilinear Bar.

Applications of Hamilton's and Lagrange's equations: A particle of mass m is moving vertically

under the action of gravity and a resistance force numerically equal to k times the displacement x

from an equilibrium position. Obtain the Hamilton's and Euler's equation.

Use Hamilton's principle to find the equations for the small vibrations of a flexible stretching string

of length I and tension T fixed at end points.

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7.

MM453A

M.Sc.(Mathematics)

Integral Equations & Calculus of VariationsPaper lllA

Practical Questions

Semester lV

1. From an lntegral equation corresponding to the differential equationyo' + xyo +(x' - x)y = xe* + l; y(0) = -y'(0) = I; /,(0) = I

Convertthedifferential equation yo'+xy" +(*, - x)y=xe, +I;withinitial conditionsy(0) =.y'(0) = l, y'(0) = 0; into volterra's lntegral Equations.

3. Solve the lntegral Equations

e'(x) + q(x) + lSinng - )(p(t)dt +'!Cosh(x - t)cp,(t)dt = Coshx ; A(0) = q,(0) = 0 .

Solve the lntegral Equations'V!)!L = x' i 0 <a <l;i(x-t)"

with the aid of Resolvent Kernel, find the solution of the lntegral equation

'-) t

e@) =e.' sin x + FI9II qQ)dti z+cost

Solve the lntegral Equations 0{x) - )"[arccost.QQ)dt =+Find the characteristic numbers uno EL"n function ,r,n"1l;*.t Equations

I

i@) * t l(+s.' togr - ot' tog x\p1t)dt = o

Applications of Green's function : Construct Green's function for the homogeneous boundary valueproblem y''' (x) = 0; y(0) = .y'(0) = 0; -v(l) = y'(l) = 0.

Applications of Green's function : solve the Boundary Value problem ./"'(.r) = l;y(0) =.rr'(0) = y'(l) = /'(0) = 0.

Apptications of Green's function : Solve the Boundary Value problem yo + y = x, ;

y(0)= y(rcl2)=0.

Find the extremals of the functionat v[.v(x)] ="{4 O*.; y

I

Test for an extremum the functionat u[y(.r)] = [G, + ),2 -21,2 y'px; y(0) = l; y(l)) = 2.0

LL.

,r.,rtiCSi,t'.,1:r',?irC::^ ll.r

L2,